Complexity and Real Computation
By: by Lenore Blum, Felipe Cucker, Michael Shub, Steve Smale
Contributors: Cucker, Felipe; Shub, Michael; Smale, Steve
Publisher: Springer New York
Published: 1998
Place of publication: New York, NY
Language: eng
Format: [electronic resource] /
Number of pages: 453
ISBN-13: 9781461268734; 9781461207016
Subjects: Mathematics; Information theory; Symbolic and mathematical Logic
Description: The classical theory of computation has its origins in the work of Goedel, Turing, Church, and Kleene and has been an extraordinarily successful framework for theoretical computer science. The thesis of this book, however, is that it provides an inadequate foundation for modern scientific computation where most of the algorithms are real number algorithms. The goal of this book is to develop a formal theory of computation which integrates major themes of the classical theory and which is more directly applicable to problems in mathematics, numerical analysis, and scientific computing. Along the way, the authors consider such fundamental problems as: * Is the Mandelbrot set decidable? * For simple quadratic maps, is the Julia set a halting set? * What is the real complexity of Newton's method? * Is there an algorithm for deciding the knapsack problem in a ploynomial number of steps? * Is the Hilbert Nullstellensatz intractable? * Is the problem of locating a real zero of a degree four polynomial intractable? * Is linear programming tractable over the reals? The book is divided into three parts: The first part provides an extensive introduction and then proves the fundamental NP-completeness theorems of Cook-Karp and their extensions to more general number fields as the real and complex numbers. The later parts of the book develop a formal theory of computation which integrates major themes of the classical theory and which is more directly applicable to problems in mathematics, numerical analysis, and scientific computing.
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Table of contents: title: Introduction; title: Definitions and First Properties of Computation; title: Computation over a Ring; title: Decision Problems and Comlexity over a Ring; title: The Class NP and NP-complete Problems; title: Integer machines; title: Algebraic Settings for the Problem "P=NP?"; title: Newton's Method; title: Fundamental Theorem of Algebra: Complexity Aspects; title: B zout's Theorem; title: Condition numbers and the loss of precision of linear equations; title: The Condition Number for Non-Linear Problems; title: The condition Number in P(H(d)); title: Complexity and the Condition Number; title: Linear Programming; title: Deterministic lower bounds; title: Probabilistic machines; title: Parallel computations; title: Some Separations of Complexity Classes; title: Weak machines; title: Additive machines; title: Non uniform complexity classes; title: Descriptive Complexity; title: Bibliography; title: Index.